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Construction of negatively curved complete intersections

Jean-Paul Mohsen

Abstract

Using the Donaldson-Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfaces and we obtain bounds for their Kobayashi hyperbolic metric.

Construction of negatively curved complete intersections

Abstract

Using the Donaldson-Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfaces and we obtain bounds for their Kobayashi hyperbolic metric.

Paper Structure

This paper contains 21 sections, 23 theorems, 84 equations.

Table of Contents

  1. Asymptotic transversality theory
  2. Limit submanifolds
  3. Jets of submanifolds
  4. Avoidance theorem
  5. Proof of Theorem \ref{['negatively curved']}
  6. Curves with negative curvature
  7. Ricci curvature
  8. Scalar curvature
  9. Holomorphic sectional curvature
  10. Holomorphic bisectional curvature
  11. Proof of Theorem \ref{['exterior']}
  12. Subbundles and stationary points
  13. Griffiths-positivity and the Serre line bundle
  14. Griffiths-positivity of $\bigwedge ^l (TY)$
  15. Proof of Theorem \ref{['hyperbo']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $X$ be a complex projective manifold of dimension $n$ equipped with a Hermitian metric $\mu$ and an ample line bundle $L \rightarrow X$.

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Proposition 9
  • Lemma 10
  • ...and 21 more